Following the PFE method, the flow is now linearised around the resulting mean flow obtained from the ideally expanded mean flow and the PSE solution for the shock-cells. In this example case, only axisymmetric disturbances (\(m=0\) ) will be analysed. The initial step of PFE requires the solution of a spatially periodic linear stability analysis truncated to the number of harmonics (\(N\) ) considered in the formulation, where the wavenumber associated with the periodicity is defined by the wavenumber of the shock-cell structure at the initial streamwise station (this is the equivalent of the spatial linear stability analysis for PSE). Figure REF  shows sample eigenspectra for the present flow at \(St=0.4\)  and \(0.7\) , where similar features to [1]} are observed. Firstly, the same modes that appear in the locally parallel case are also evident here; however, these modes now appear several times, with repeated modes offset from each other by \(\alpha _s\) . This is exemplified by the evanescent acoustic branch: for both frequencies, this branch arises at \(Re(\alpha )=0\)  and \(Re(\alpha )=\alpha _s = 11.6\) , with the equivalent occurring for the discrete upstream mode and the KH mode. As mentioned in section REF , this repetition of the spectrum leads to the stable/evanescent upstream modes being interpreted by the spatial march as unstable downstream-travelling modes. Frequencies where the KH mode is close to the guided jet mode are also expected to be close to the screech frequency, as shown in [1]}; due to the presence of both downstream- and upstream-travelling waves in the absolute instability phenomenon, the current PFE method is not appropriate to analyse the resulting flow structures associated with resonance arising from absolute instability. Thus, these frequencies will be avoided in the present analysis.
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